# Visualizing 4D space

AlchemistK
I've been having trouble visualizing 4D objects(4 physical dimensions, excluding time), in fact I can't seem to be able to do it at all.Can someone help me on it?

I don't simply want to imagine how they might simply appear in 3D space and show their 3D projections as they pass by. I want to be able to rotate my objects!

Is that even possible? I did try googling it, but the results weren't satisfactory.

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## Answers and Replies

_PJ_
There's no way to realize any visualisation of 4D pobjects unless you restirct the view to a 'slice' of the fourth dimension, or, as a 3D shadow of a 4D object.

It's difficult to imagine greater than 3 dimensions for anyone, since we have no frame of reference beyond our typical 3 and there's no means to perceive such dimensionality.

AlchemistK
I was watching these videos on dimensions, really amazing, and they used this technique of inflating 3D objects and then projecting the image in 2 dimensions. Then they showed 4D objects in a similar way. How did they think of those up? And how were the figures of "Hypercubes" and other 4D objects made?

_PJ_
"Hypercubes" are extra-dimensional cubes. A Tesseract is a Hypercube of 4 dimensions.
The Tesseract is often displayed as a 3D object which represents and is theoretically identical to the "shadow" of a 4D cube.

The reasoning behind this use of shadows, is that, as a shadow of a 3D object becoimes visible as a distorted 2D image, so too, is a 3D distorted 'image'(though in this case the 'image' will have 3 dimensions) considered like the "shadow" of a 4D object.

The distortion of the shadow image "flattens" the original object and therefore there is a loss of information as the perspective in the higher dimension is erased.

To give an analogous example, consider a 2D shadow of a 3D cube:

.___.
/__/ |
| | |
L_| /

In a real, 3D cube, every edge has the same length. Every vertex is 90 degrees.
However, in the (poorly drawn) 2D shadow above, there are non-right-angles and some sides longer/shorter, but without this distortion, the cube would be impossible to draw in 2D.

Conventional depictions of Tesseracts rely on the same principle. By distorting the apparent lengths and angles (which, in any dimensional cube would still be all the same length and all 90 degrees), the 4D objects can be shown as 3D and lower 'shadows'

Dremmer
In order to realistic visualize a 4D object, you would have to visit a four-dimensional space. At least presently, such is not possible. Any image of a 4D object in a three-dimensional space will really be 3D.

pointlesslife
You actually see the world in 2D and the brain creates an 3D illusion by comparing images from both eyes.

Imagining a 4D object? You can try by looking at a tesseract with stereoscopic vision. GL with that.

http://dogfeathers.com/java/hypercube2.html

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AlchemistK
OK. But I still don't understand why these figures are as they are. Did mathematicians follow some algorithm to make them?

James_Harford
You actually see the world in 2D and the brain creates an 3D illusion by comparing images from both eyes.

Imagining a 4D object? You can try by looking at a tesseract with stereoscopic vision. GL with that.

http://dogfeathers.com/java/hypercube2.html

This is probably as good a solution as one can expect. If we lived in a 4-dimensional space, and if our eyes followed the same Iris-lens-retina design as in three dimensions, the shape of an eyeball would that of a hypersphere consisting of a 4-dimensional hypervolume enclosed by a 3-dimensional "surface".

The retina would therefore be a 3-dimensional volume, into which 3-dimensional perspective images would be projected. Therefore the above stereo images viewable with 3-D glasses represent what a one-eyed inhabitant of this 4-D world would see. As one-eyed people are not overly confused by the three-dimensional world, so it seems that in principle we too should can develop a 4-dimensional imagination.

On the other hand it may be that, as with language, this sense can be developed only in early childhood. In which case, one should start in, say, pre-school?

Gold Member
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Flustered
What are experiments that could be ran, to detect extra dimensions?
How does one go about looking for a 4th dimension.

Alephu5
I cannot visualise looking at an nD shape, but if you want to imagine being in a hyperdimensional universe it's pretty simple. Basically just extrapolate from the experience of repeatedly turning through 90° in a 3D universe; on the fourth turn you see the same view as when you started turning. In an nD universe, you would have to turn n+1 times before you saw the same view as when you started, so in 4D space you need to turn 5 times.

James_Harford
I cannot visualise looking at an nD shape, but if you want to imagine being in a hyperdimensional universe it's pretty simple. Basically just extrapolate from the experience of repeatedly turning through 90° in a 3D universe; on the fourth turn you see the same view as when you started turning. In an nD universe, you would have to turn n+1 times before you saw the same view as when you started, so in 4D space you need to turn 5 times.

Seeing the original view on the fourth turn is true in 2-d space if each rotation is in the same plane.